1. Statistical Mechanics and Multi-Scale Simulation Methods ChBE 591-009
Prof. C. Heath Turner
Lecture 05
Some materials adapted from Prof. Keith E. Gubbins:
Some materials adapted from Prof. David Kofke:

2. Open-Shell Systems
Spin-restricted HF (RHF) = uses same spatial functions for both the a and b spins.
Spin-unrestricted HF (UHF) = uses separate spatial functions for the a and b spins, resulting in two Fock matrices:
** electron spin everywhere is NO LONGER zero. A spin density can now be defined:
When is UHF needed?
open-shell systems
radicals
molecules undergoing dissociation
some ground-state molecules

3. Open-Shell Systems
Spin-unrestricted HF (UHF) - visualization

4. Advanced Ab Initio Methods
Configuration Interaction
Møller-Plesset perturbation theory
Coupled-Cluster Theory
Gaussian-2/Gaussian-3
ONIOM

5. Advanced Ab Initio Methods
Electron Correlation
HF: e- move in an average potential. Dynamics of e- neighbors (called ‘dynamical correlation’) are neglected.
Fock operator = 1e- operator  1e- MOs  wave function (Slater determinant)
Ecorr = E – EHF
Example: H2  H + H, w/o e- correlation e- are predicted to spend equal time on both nuclei, even at infinite H-H separation.
wavefunction is a single determinant – sometimes too rigid…
Example: Degenerate Orbitals
Alternate approach: construct the wavefunction using multiple determinants (ci = weight factors, subject to normalization)

6. Beyond HF: Configuration Interaction
The CI method employs a wavefunction which is constructed of a
linear combination of the HF wavefunction with excited determinants
The expansion coefficients are then selected so that they
variationally minimize the expectation value of the electronic energy
with respect to the CI wavefunction
where the wavefunction is restricted by the normalization condition

7. Configuration Interaction for Singles
If we imagine the case of H2 we can implement CI for
singles by mixing in the excited state configurations. In
molecules of high symmetry only the configurations of
appropriate symmetry
can “mix” in this way.
CI that includes singles
only is appropriate for
improving transition
energies, but does
not help ground
state properties.

8. Multielectron Configuration Interaction
The CI expansion is variational and, if the expansion is complete (Full CI), gives the exact correlation energy (within the basis set approximation).
The number of determinants in Full CI grows exponentially with the system size, making the method impractical for all but the smallest systems. For this reason the CI expansion is usually truncated at some order, for example CISD, where only singly and doubly excited determinants are considered.
Brillouin's Theorem states that singly excited determinants do not mix with the HF determinant. Therefore CISD is the cheapest worthwhile form of CI, yet this method scales as O(N6) where N is the size of the system.
CI is NOT size consistent: (EA+EB)CI ≠ (EA)CI+(EB)CI
QCISD = size-consistent CISD theory

9. Perturbation Theory Møller-Plesset perturbation theory (MPn)
** Used to account for e- correlation:
Create a more tractable operator.
Using exact eigenfunctions and eigenvalues of simplified operator, estimate the eigenfunctions and eigenvalues of the more complete operator.
Complete operator = H (Hamiltonian):
Correction term to the approximate Hamiltonian = difference b/t true Hamiltonian (1st term) and Fock operator (2nd term):
The wave function and corresponding energy can be found:

10. Perturbation Theory Møller-Plesset perturbation theory (MPn)
The various energy values can be calculated if the wavefunctions are known:
** The higher order wave functions developed by promoting electrons into the virtual orbitals taken from the HF calculation.
MP2 = double excitations
MP3 = little improvement over MP2
MPn = size independent, not variational

11. Other Methods…
Coupled-cluster (CC) theory – a perturbation theory of e- correlation with an excited configuration that is “coupled” to the reference configuration.
CCD = include double excitations
CCSD = single + double excitations
CCSD(T) = includes triples (approximately)
Gaussian-2/Gaussian-3 (G2/G3) = composite methods using various levels of theory to compute thermochemistry
ONIOM = model critical parts of the system with various levels of theory

12. Density Functional Theory
Increased in popularity within last 2 decades.
Given a known e- density  form the H operator  solve the Schrödinger Eq.  determine the wavefunctions and energy eigenvalues.
HF – the wavefunction is essentially uninterpretable, lack of intuition.
Hamiltonian depends ONLY on the positions and atomic number of the nuclei and the number of e-.
HF – optimize the e- wavefunction
DFT – optimize the e- density

13. Definitions Function: a prescription which maps one or more numbers
to another number: y = f(x) = x2.
Operator: a prescription which maps a function onto another
function:
Functional: A functional takes a function as input and gives
a number as output. An example is:
F[f(x)] = y
Here f(x) is a function and y is a number.
An example is the functional to integrate x from –¥ to ¥.